OF BLACKFOLD KENTARO TANABE (UNIVERSITY OF BARCELONA) - - PowerPoint PPT Presentation

SELF-GRAVITY EFFECTS OF BLACKFOLD

KENTARO TANABE

(UNIVERSITY OF BARCELONA)

collaboration with R. Emparan and S. Kinoshita

• 1. INTRODUCTION

blackfold approach effective worldvolume theory for dynamics of black brane Application

𝑆 ≫ 𝑠0

𝑆

• construction of higher dimensional black holes
• stability analysis of black brane
• D-brane probe in thermal background

curvature effects ~ 𝑠

𝑆

[Emparan et.al. (2007,2009)] [Camps, et.al. (2009)] [Grignani, et.al. (2011), Arms, et.al. (2012),..]

BLACKFOLD AS BLACK HOLE

black hole solution can be constructed by gluing branes various properties of higher dimensional black holes

1. possible topology of black holes 2. phase diagram of solutions,…

Matched Asymptotic Expansion

PRESENT STATUS

possible topology phase diagram Note: these studies were done by test brane analysis

PURPOSE

Our purpose:

• clarify its matching structure
• computing back reaction (self-gravity

effects)

how solution is constructed by blackfold approach check the validity of blackfold approach construction of more precise phase diagram

• 2. MATCHING LADDER

BLACK RING AS BLACKFOLD

Black ring is constructed from black string ( black 1-brane )

boosted black string bending

black string + perturbation flat spacetime + perturbation match

𝑠

black ring

𝑆

near region far region

schematic matched asymptotic expansion

BLACKFOLD EQUATION

𝑈𝜈𝜉𝐿

𝜈𝜉 𝜍 = 0

brane should satisfy regularity condition ( “no tension” )

brane’s energy momentum tensor extrinsic curvature

This condition guarantees the regularity of the black hole horizon and determines possible topology

[Camps and Emparan (2011)]

FIRST MATCHING

First order solution at far region (Newton solution) 𝑆

boosted black string constitute the 𝜀 – function source at far region

𝑈𝜈𝜉𝐿𝜈𝜉

𝜍 = 0

MATCHIG AT NEAR REGION

bending the black string excites the perturbation on the black string

=

black string

+

perturbation

𝑕𝜈𝜉 𝐿

𝜈𝜉 𝜍 𝑧𝜍

𝑧𝜍

From dimensional analysis,

𝑚 mode perturbation

= 𝐵𝑚 𝑠 𝑆

𝑚

Amplitude is determined by far region solution

𝑚 : spherical harmonics

MATCHING AT FAR REGION

perturbation on black string also excite new energy momentum tensor at far region

𝐵𝑚 𝑠 𝑆

𝑚

1 − 𝑠0

𝑒−4

𝑠𝑒−4

excited by far region solution create new energy momentum tensor at far region

MATCHING LADDER

far region = flat + perturbation near region = black string + perturbation

black string Newton solution

𝑈𝜈𝜉𝐿

𝜈𝜉 𝜍 = 0

𝑚 mode perturbation 𝑠 𝑆

𝑚

correction to energy momentum tensor 𝑠 𝑆

𝑚

( at least )

3.SELF-GRAVITY EFFECTS

SELF-GRAVITY

using the matching ladder, compute the self-gravity effects self-gravity order

∆ Ψ(2) = Ψ2

self-gravity correction Newton solution

Ψ 2 = 𝑃 𝑠 𝑆

𝑒−4

A B

SECOND ORDER

For example, consider 6-dimensions case

black string Newton solution

𝑈𝜈𝜉𝐿

𝜈𝜉 𝜍 = 0

self-gravity correction

𝑠 𝑆

2

𝑚 = 1 mode

𝑠 𝑆

1

𝑚 = 0 mode

𝑠 𝑆

2

corrections to mass, angular momentum and area

CORRECTIONS

Solving and matching the perturbations, the self-gravity effects shows up to trivial changing ( 1st law of black hole ) 𝑁 = 𝜌 16𝐻 𝑆𝑠

2

𝐾 = 𝜌 4𝐻 𝑆2𝑠

2

𝑇 = 𝜌 6 𝑆𝑠

3 1 + 𝐷1 + 1

4 log 𝑠 𝑆 𝑠 𝑆

2

mass angular momentum

area self-gravity correction

( 𝐷1 < 0 )

PHASE DIAGRAM

Phase diagram is useful to see the physical effects of self-gravity

1st order solution 2nd order solution (self-gravity ) 𝑘𝑒−3 = 𝑘0 𝐾𝑒−3 𝑁𝑒−2 𝑡𝑒−3 = 𝑡0 𝑇𝑒−3 𝑁𝑒−2

• 4. SUMMARY
• complete the matching ladder of blackfold

approach (black string case)

• compute the self-gravity effects

future work

• matching ladder of general blackfold
• incorporating the intrinsic perturbation